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A sound card based multi-channel frequency measurement system
S. Groeger1,2, G. Bison1, P.E. Knowles1,a, and A. Weis1
1 Physics Department, Universit´e de Fribourg, Chemin du Mus´ee 3, 1700 Fribourg, Switzerland
2 Paul Scherrer Institute, 5232 Villigen PSI, Switzerland
Abstract. For physical processes which express themselves as a frequency, for example magnetic field measurements using optically-pumped alkali-vapor magnetometers, the precise extraction of the frequency from the noisy signal is a classical problem. We describe herein a frequency measurement system based on an inexpensive commercially available computer sound card coupled with a software single-tone estimator which reaches Cram´er–Rao limited performance, a feature which commercial frequency counters often lack.
Characterization of the system and examples of its successful application to magnetometry are presented.
PACS. 06.30.Ft Time and frequency – 07.05.Hd Data acquisition: hardware and software – 07.55.Ge Magnetometers for magnetic field measurements
1 Introduction
The present work is motivated by the need for a high res- olution frequency measurement system for analyzing sig- nals generated by optically-pumped cesium magnetome- ters [1]. A set of such magnetometers will be used for a detailed investigation of magnetic field fluctuations and gradients in an experiment searching for a neutron elec- tric dipole moment (nEDM). The experiment calls for a magnetic field of between 1 to 2μT controlled at the 80 fT level when measured over 100 s time intervals, control cor- responding to a relative uncertainty between 40 to 80 ppb.
The magnetometers are based on the fact that for low magnetic fields the Larmor precession frequency fL in a vapor of Cs atoms is proportional to the modulus of the magnetic fieldB
fL=γ|B|. (1)
The proportionality factor γ is a combination of funda- mental and material constants and has a value of about 3.5 kHz/μT for133Cs. The precession of the atoms mod- ulates the resonant absorption coefficient of the cesium vapor, which is measured by a photodiode monitoring the power of a laser beam traversing the atomic vapor [2].
In the self-oscillating mode of operation [2,3] the magne- tometer signal is of the form
s(t) =Asin (2πfLt+φ) +s0. (2) The Larmor frequency, fL, has to be extracted from the signal. Equation (1) connects the frequency determination
a e-mail:Paul.Knowles@unifr.ch
precision directly to the resulting field measurement pre- cision. The basic demand on the frequency measurement system in order to achieve the required field precision is a resolution of a few hundredμHz in an integration time of 100 s. Moreover, the synchronous detection of signals from an array of magnetometers requires a cost-effective multi-channel solution.
In our recent study [3] of optically–pumped magne- tometer performance, frequency measurements were made with a commercial frequency counter (Stanford Research Systems, model SR620), which has a limited frequency res- olution, thereby limiting the magnetic field determination.
Frequency counters rely on the detection of zero crossings of a periodic signal in a given dwell time. Their perfor- mance is limited by their resolution of the zero crossing times, an event which is affected by the amplitude, offset, and phase noise of the signal. In demanding applications, such as the one investigated here, that timing jitter limits the ultimate frequency resolution of the magnetometer sig- nal measurement. Put simple, the limitation of frequency counters is due to the fact that they use only informa- tion in the vicinity of the zero crossings, while valuable wave form information from in between the zero crossings is ignored.
As a more powerful alternative one can use numerical frequency estimation algorithms to extract the frequency from the complete waveform sampled at an appropriate rate and with a sufficient resolution. The performance of an ADC-based measurement system for measuring a sin- gle frequency of about 8 Hz was discussed in [4]. Under the assumption that a stable clock triggers the ADC, the
Published in "The European Physical Journal - Applied Physics 33: 221-224, 2006"
which should be cited to refer to this work.
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authors in [4] show that the lower limit of the frequency resolution of their system coincides with the Cram´er–Rao lower bound (CRLB) [5]. The CRLB is a well-known con- cept from information theory and describes principle lim- its for the estimation of parameters from sampled signals.
In our application, the Larmor frequency in a mag- netic field of 2μT lies in the audio frequency range (fL = 7 kHz). We have investigated whether a com- mercially available (and rather inexpensive) professional multi-channel sound card would present a viable solution for sampling the magnetometer signals. The estimation of the frequency from the sampled data was done by a soft- ware algorithm. In the following we will show that such a simple system can indeed be used for CRLB limited real- time frequency measurements and for a detailed study of noise processes which limit the precision of atomic mag- netometers.
2 The system
The frequency measurement system consists of a profes- sional sound card (M–Audio Delta 1010) for digitizing the analog input data, an atomic clock to provide a stable time reference, and a standard personal computer (PC) which reads the data and runs the frequency estimation algorithm. The sound card provides 8 analog input chan- nels in a breakout box that connects to a PCI interface card in the PC. The analog input signals can be sampled with a resolution of up to 24 bit at a sampling rate of up to 96 kHz. In order to limit the amount of data we used only 16-bit resolution, which was proven to yield suffi- cient precision. Jitter or drifts of the sampling rate induce additional phase noise on the sampled signal which can seriously degrade the precision of the frequency estima- tion. An essential feature of the Delta 1010 sound card is its “world clock” input which can be used to phase- lock the internal clock of the sound card to an external 96 kHz time base. The time base was realized by a fre- quency generator synchronized to the 10 MHz signal of a rubidium frequency standard (Stanford Research Sys- tems, model PRS10). The Rb frequency standard provides a relative stability of 10−12in 100 s which minimizes pos- sible sampling rate jitter and drifts far below the required level. The requirements for the PC system are not very de- manding as long as it allows for the continuous recording of the 16 bit data sampled at a rate of 96 kHz (5.8 GB/h for 8 channels). A 1.8 GHz Pentium-4 processor was fast enough for real-time frequency determination for all eight channels at a given integration time. However, for the de- tailed analysis described below, in which the integration time is varied, the time series were evaluated off-line from the stored sampled data.
3 Performance
Considering the magnetometer signal given by equa- tion (2), the frequency fL is to be determined from the
AC-coupled signal data which, after sampling, are of the form
xn =Asin
2πn
k=1
(fL+δfk)Δt+φ0+δφn
+δxn,(3) n= (0, . . . , N−1),
where A is the signal amplitude, Δt the time resolution (inverse of the sampling raters), andφ0 the initial phase.
The number of sample points isN =τ/Δt(=τrs), where τ is the measurement integration time. Also shown is the noise contribution at each pointnarising from phase noise δφn, frequency noisen
k=1δfk, and offset noiseδxn. The frequency is determined from the dataxn by a maximum likelihood estimator based on a numerical Fourier trans- formation which provides a CRLB limited value [6]. The algorithm iteratively searches for the frequency f that maximizes the modulus of the Fourier sum
MF(f) =
N−1 n=0
xnWnexp
i2πf rs n
, (4)
where Wn is a windowing function.
Under ideal conditions (stable field and ideal electron- ics), the frequency and phase noise (δfk andδφn) are not present in the signal (Eq. (3)). The fundamental noise contribution is the photocurrent shot noise, which is pro- portional to the square root of the DC offset s0 in equa- tion (2). The noise is converted, by a transimpedance amplifier, to voltage Vpc and has a Gaussian amplitude distribution with zero mean, corresponding to a white frequency spectrum that is characterized by its power spectral density ρ2x (in V2/Hz). The signal-to-noise ratio (SNR) is defined as A2/(ρ2xfs), where fs = (2Δt)−1 = rs/2 is the sampling rate limited bandwidth, i.e., the Nyquist frequency or the highest frequency that can be detected unaliased. The CRLB of the frequency estima- tion from such an ideal magnetometer signal is given by the variance [5]
σCRLB2 = 3ρ2x
π2A2τ3. (5) In frequency metrology it is customary to represent fre- quency fluctuations in terms of the Allan standard devia- tionσA – or its squareσ2A, the Allan variance [7,8]. One can show that for white noise σA coincides with the clas- sical standard deviation [9]. A double logarithmic plot of the dependence ofσA on the integration timeτ is a valu- able tool for assigning the origin of the noise processes that limit the performance of an oscillator (see for ex- ample [7,8]). As shown in Table 1, the variance σA2 de- pends both on integration timeτand measurement band- widthfc, which, for a measurement intervalτ, is given by fc= (2τ)−1. When that relation between bandwidth and integration time is inserted into the formulas given in the central column of Table 1 [8], one finds the typical τ de- pendencies of the Allan standard deviation σA shown in the right-hand column. In the presence of several uncor- related noise processes, α, the variance of the estimated
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0.1 1 10 100
10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2
Integration timet(s) t-3/2
t-3/2 t-1 t-1/2 t0
Allanstd.dev.(Hz)sA
Fig. 1.Allan standard deviation of the frequency of a synthe- sized sine wave affected by different noise processes. From top to bottom: flicker frequency noise (black dots), white frequency noise (open circles), flicker phase noise (black triangles), white phase noise (open triangles), white offset noise (black squares).
Table 1.The central column shows the dependence of the Al- lan variance σ2A on the integration time τ and measurement bandwidthfcfor the noise sources listed at the left [8]. White noise sourcesαare characterized by their power spectral den- sity ρ2α. The frequency dependent spectral density of flicker noise processαis modeled byρ2α(f) =h2α/f. By assuming the relationfc= (2τ)−1we find the power laws which typify each noise type, shown in the right hand column.
noise source σA2(τ) σA(τ)
white offset 3
π2A2·ρ2x·1
τ3 ∝τ−3/2 flicker frequency 2 ln 2·h2f·1 ∝τ0 white frequency 1
2·ρ2f·1
τ ∝τ−1/2
flicker phase 3
4π2·h2φ·ln(2πfcτ)
τ2 ∝τ−1
white phase 3
8π·ρ2φ·fc
τ2 ∝τ−3/2
frequency is given by
σ2=
α
σα2(f). (6)
Note that for a magnetometer signal, the contribution from equation (5) will always be present in the sum.
We first investigated whether our data analysis algo- rithm reproduces the theoreticalτ-dependencies shown in Table 1. For that purpose we generated time series (16 bit, 96 kHz) corresponding to equation (3) with only one of the phase, frequency, or offset noise terms enabled, and se- lected with well defined spectral characteristics (flicker or white). Figure 1 shows the Allan standard deviationσAof those synthetic data. The emphasis here lies on the slopes rather than on the absolute values, which were chosen to yield a readable graph.
Integration timet(s)
Allanstd.dev.(Hz)sA Relativefrequencyuncertainty
0.01 0.1 1 10 100 1000
10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5
t-3/2 t-1
e) b)
c) d)
sound card
frequency counter
a)
Fig. 2. Allan standard deviation of the frequency of a sine wave affected by different noise processes, measured with the sound card (a–c, e) and a frequency counter (d): (a) white offset noise, (b) white offset noise and flicker phase noise, (c) white offset noise and flicker frequency noise. The dashed lines rep- resent the dependencies calculated on an absolute scale us- ing the applied noise amplitudes. (d) The same signal as in (a) measured with a commercial frequency counter (Stanford Research model SR620) with a 300 Hz input bandpass filter.
(e) Sound card measurement of the cleanest signal available from the function generator.
Next, we investigated the ability of the sound card to reach CRLB limited detection of a 7 kHz sine wave. The wave was generated by a digital function generator (Agi- lent, model 33220A) stabilized to the same Rb frequency standard as the sound card. In order to simulate a signal comparable to that of the magnetometers, the SNRA/ρx
of the function generator output was artificially decreased from its nominal value of better than 5×105√
Hz to about 1.3×105√
Hz (in 1 Hz bandwidth) by adding white offset noise. We recorded a 1 h time series of that signal, sampled with 16-bit resolution. The data were analyzed with the same algorithm as above and yielded an Allan standard deviation σA(τ), shown as black dots in Figure 2a. The measurement agrees on an absolute scale with the CRLB calculated using equation (5) and the applied SNR. In ad- dition to the offset noise, a second noise source was used to apply 1/f noise, in turn, to the frequency or to the phase modulation input of the function generator. The result- ing σA of the measured data is shown in Figures 2b, c.
Figure 2d shows σA derived from the same signal as Fig- ure 2a but analyzed by the commercial frequency counter (Stanford Research Systems, model SR620) that was used in [3]. The three points shown correspond to the three possible integration times of the SR620. It can be clearly seen that the counter technique does not allow the correct measurement of these faint noise processes. However, ex- trapolation of the data points suggest that for integration times less than 10 ms the CRLB could be reached.
Figure 2e showsσAderived from a measurement of the cleanest signal the function generator could produce. For integration times above a few seconds, the measured Allan standard deviation no longer follows the calculated CRLB.
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Integration timet(s) Allanstd.dev.(Hz)sA
0.01 0.1 1 10 100 1000
10-7 10-6 10-5 10-4 10-3 10-2
10-11 10-10 10-9 10-8 10-7 10-6
sAL/f b)
a)
c)
Fig. 3. (a) ASD σA of magnetic field fluctuations inside a multi-layer magnetic shield. (b) Residual fluctuations of the stabilized magnetic field. The dashed lines indicate the CRLB (left) and an assumed white frequency noise limitation (right).
(c) Stability required for the proposed nEDM experiment.
This measurement indicates some kind of lower limit of the frequency resolution of the sound card for integration times above a few seconds. We could not clarify if that behavior arose from a limitation within the sound card, or from elsewhere in the signal treatment chain.
Finally, after the frequency estimator algorithm and the sound card had proven their CRLB performance limit, we used the system to analyze the frequency generated by an optically-pumped magnetometer (OPM). A magnetic field of 2μT was produced by a solenoid driven by an ultra-stable current source. The OPM signal in that field is a 7 kHz sine wave. The OPM and the solenoid were located in a 6-layer magnetic shield in order to suppress external field fluctuations. Figure 3a shows σA of a 2 h time series recorded with the sound card. The data rep- resent pure magnetic field fluctuations. In particular, the approximately 2 mHz fluctuations in the range between 2 to 200 s could be traced back to irregular current fluc- tuations in the solenoid. Nevertheless, the relative field stability – and therefore the relative current stability – is on the order of 3×10−7 for that range of integration times. However, for a 100 s integration time the field insta- bility exceeds the requirement for the nEDM experiment mentioned in the introduction.
In order to determine the magnetometer performance limit, we actively stabilized the magnetic field in the fol- lowing way. The magnetometer frequency was compared to a stable reference oscillator (i.e., the Rb frequency standard) by means of a phase comparator, and the error
signal was used to control the solenoid current, thus real- izing a phase-locked loop. Figure 3b shows the Allan stan- dard deviation of the OPM in the stabilized field, which is CRLB-limited up to an integration time of 1 s. The noise excess between 1 and 300 s above the limits expected from the CRLB and the assumed white noise limitation shows the limitation of the current stabilization scheme, which nonetheless allows the suppression of the fluctuations by three orders of magnitude at the integration time of inter- est.
We have realized a frequency measurement system based on a digital sound card and have shown that it yields a performance superior to commercial frequency counters.
We have proven that the system yields CRLB limited fre- quency resolution in measurements of sine waves affected by various sources of noise. We have used the system to prove that, at least in a limited range of integration times, an active field stabilization by an optically pumped magnetometer is limited by the theoretical Cram´er-Rao bound. The performance and the multi-channel feature of the sound card and its external frequency reference option present a low-cost alternative for applications requiring si- multaneous characterization of several frequency genera- tion systems, especially for long integration times.
We thank Francis Bourqui for help in developing the read- out software. We acknowledge financial support from Schweiz- erischer Nationalfonds (grant No. 200020–103864), and Paul Scherrer Institute (PSI).
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